Answer
$15$
Work Step by Step
Given: $r(t)=12ti+8t^{3/2}j+3t^2k$ ; $0 \leq t \leq 1$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt 12,12t^{1/2},6t\gt$
and $|r'(t)|=\sqrt {( 12)^2+(12t^{1/2})^2+(6t)^2}dt$
$=12+6t$
$L=\int_{0}^1(12+6t) dt$
$\implies L=12t+3t^2|_{0}^1$
$=12(1)-12(0)+3(1)-3(0)$
$=15$
Hence, $L=15$
